Hydrogen energy levels Eₙ = −13.6 - n² eV: Visual walkthrough

The cast of characters (defined before we use them)

Look at the picture: the proton (magenta dot) sits still, the electron (violet dot) circles it at distance $r$ , moving with speed $v$ along the orange arrow. That is the entire physical system. Now we ask: what holds the electron in that circle, and how much energy does it have?

Step 1 — What pulls the electron in? (Coulomb's law)

WHAT. Opposite charges attract. The proton ( $+ e$ ) pulls the electron ( $- e$ ) straight toward the centre with a force we call $F_{pull}$ .

WHY this tool. We need to know how strong the pull is, because that strength decides how fast and how far the electron can orbit. Coulomb's law is exactly the rule that answers "how strong is the electric pull between two charges a distance $r$ apart?"

PICTURE. The force points from electron straight to proton — always inward, along the radius.

Step 2 — Why does that pull make a circle? (Centripetal force)

WHAT. Anything moving in a circle is constantly being turned toward the centre. The exact amount of inward force needed to bend a mass $m$ moving at speed $v$ into a circle of radius $r$ is the centripetal force, $F_{turn} = \frac{m v ^{2}}{r}$ .

WHY this tool. We have a pull (Step 1) and we see a circle (Step 1 picture). Centripetal force is the bridge: it tells us exactly how much inward force a circle demands. If the pull supplies precisely that demand, the circle is stable.

PICTURE. The inward pull arrow and the "required turning" arrow are the same arrow. That coincidence is what makes a circle possible.

Step 3 — The quantum rule: only certain circles are allowed

WHAT. So far any radius works — classical physics allows every circle. But real atoms don't do that; they use only special sizes. Bohr's rule (from the Bohr model of the atom) picks the survivors: the electron's angular momentum $m v r$ must be a whole-number multiple of a tiny fixed amount $ℏ$ .

WHY this tool. Without a rule to forbid most orbits, the electron would spiral inward and the atom would collapse. We need something that says "step here, and here, but nowhere between." Quantization of angular momentum (a quantum number $n$ ) is that filter.

PICTURE. Think of a guitar string: only whole numbers of waves fit around the loop. $n = 1$ is one loop, $n = 2$ is two, and nothing in between resonates.

Step 4 — Pin down the radius $r_{n}$

WHAT. We now have two facts about the same orbit: the force balance $(⋆)$ and the quantum speed $v = \frac{n ℏ}{m r}$ . Feed the second into the first and the unknown $r$ gets locked to a specific value for each $n$ .

WHY this step. Two equations, two unknowns ( $v$ and $r$ ). Substituting kills $v$ and leaves $r$ alone — algebra's way of saying "these two rules together allow only one radius per shelf."

PICTURE. The allowed circles are not evenly spaced — they balloon outward as $n^{2}$ .

Step 5 — Add up the energy (kinetic + potential)

WHAT. The electron's total energy $E$ has two parts: kinetic (energy of motion, always positive) and potential (stored energy of the attraction, negative because the charges want to fall together).

WHY this step. "Energy level" means total energy. We must combine both pieces to get it — and the [!mistake] in the parent note (thinking $E$ is positive) comes exactly from forgetting the potential piece.

PICTURE. A bar chart: KE points up, PE points down twice as far, and their sum lands below zero.

Step 6 — Insert $r_{n}$ → the famous $- 13.6/ n^{2}$

WHAT. We have energy in terms of $r$ (Step 5) and $r$ in terms of $n$ (Step 4). Plug one into the other and every symbol except $n$ collapses into a single number.

WHY this step. This is the payoff: substituting $r = a_{0} n^{2}$ turns "energy of an orbit" into "energy of shelf $n$ " — a formula in $n$ alone.

PICTURE. The ladder of levels: deep, lonely $n = 1$ at the bottom; the rungs bunch together just below $E = 0$ as $n \to \infty$ .

Step 7 — The edge cases (never leave a scenario unshown)

WHAT & WHY. A good derivation must survive its extremes. Let's push $n$ to its limits.

The figure shows the whole run: the deep floor at $n = 1$ , rungs crowding upward, the $E = 0$ ceiling they approach but never cross, and the forbidden $n = 0$ region marked off.

The one-picture summary

Here is the entire chain compressed into a single flow — you can rebuild the whole derivation from this one image.

Recall Feynman retelling — the whole walkthrough in plain words

A proton and an electron are alone in space. The proton tugs the electron toward it (Step 1, Coulomb) — and that tug is exactly the sideways push needed to bend the electron into a circle (Step 2). If nature stopped there, the electron could circle at any size and would spiral in and crash. So nature adds a rule (Step 3, Bohr): only circles that fit a whole number of quantum "loops" survive, labelled $n = 1, 2, 3 \dots$ . Combining the tug-rule with the loop-rule fixes the size of every allowed circle — they grow like $n^{2}$ (Step 4). Now add up the electron's energy: its motion energy is positive, but the fall-together energy is twice as negative, so the total always ends up negative — the electron is stuck in a well (Step 5). Swapping in the fixed circle sizes turns that total into one clean formula, $- 13.6/ n^{2}$ eV, where $13.6$ is just a bundle of nature's constants (Step 6). The lowest shelf $n = 1$ is deepest ( $- 13.6$ eV, costing $13.6$ eV to escape); the shelves crowd toward $0$ as you climb, and there is simply no $n = 0$ (Step 7). That staircase is the atom.

Connections

Parent: Hydrogen energy levels
Coulomb's law — the inward pull (Step 1)
Centripetal force — why the pull makes a circle (Step 2)
Bohr model of the atom · Quantum numbers — the quantization rule (Step 3)
Rydberg constant and formula — where $13.6$ eV comes from (Step 6)
Ionization energy — the $n = 1 \to \infty$ cost (Step 7)
Hydrogen spectral series · Photon energy E = hf = hc/λ — what the levels produce